Abstract
Lewis’s counterfactual logics are a class of conditional logics that are defined as extensions of classical propositional logic with a two-place modal operator expressing conditionality. Labelled proof systems are proposed here that capture in a modular way Burgess’s preferential conditional logic \( \mathbb {PCL}\), Lewis’s counterfactual logic \( \mathbb {V}\), and their extensions. The calculi are based on preferential models, a uniform semantics for conditional logics introduced by Lewis. The calculi are analytic, and their completeness is proved by means of countermodel construction. Due to termination in root-first proof search, the calculi also provide a decision procedure for the logics.
This work was partially supported by the Academy of Finland research project no. 1308664 and by the project TICAMORE ANR-16-CE91-0002-01.
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Notes
- 1.
The definition can be extended to the propositional formulas of the language in the standard way [17].
- 2.
The proofs of termination and completeness for systems with Uniformity and Absoluteness can be given adopting the reformulation of the calculi from Remark 1. The proofs for the current versions of the calculi would be unnecessarily complex.
- 3.
The saturation conditions for the other propositional rules are standard [20].
- 4.
Observe that \( \mathsf {Repl}\) does not introduce new labels; however, it could introduce new links between the nodes of the graph. In the presence of \( \mathsf {Repl}\) the structure generated by R is a graph; otherwise, it is a tree.
- 5.
In case of centering it is convenient to define worlds as equivalence classes, to account for formulas \( x=y \). Thus, \( [x] = \{ y \mid x=y \text { occurs in } \downarrow \varGamma \} \) and \( W^c = \{[x] \mid y \text { occurs in } \downarrow \varGamma \cup \downarrow \varDelta \} \). Centering follows from the saturation condition (\( \mathsf {C}\)).
- 6.
In case of centering, we also need to show that if \( [x] \vDash _\rho A \) and \( y \in [x] \), then \( [y] \vDash _\rho A \), and that if \( [x] \vDash _\rho A \) then x : A occurs in \( \downarrow \varGamma \). The proof follows from admissibility of \( \mathsf {Repl}\) in its generalized form [20].
- 7.
Refer to [7] for complexity results for conditional logics.
- 8.
The Limit Assumption states that there are no infinite descending \( \leqslant _x \)-chains.
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Girlando, M., Negri, S., Sbardolini, G. (2019). Uniform Labelled Calculi for Conditional and Counterfactual Logics. In: Iemhoff, R., Moortgat, M., de Queiroz, R. (eds) Logic, Language, Information, and Computation. WoLLIC 2019. Lecture Notes in Computer Science(), vol 11541. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-59533-6_16
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